Administration officials have emphasised that Trump remains committed to meeting with Kim, but they have to nail down key details for the summit, including the agenda and security logistics.
Trump appeared to suggest Kim could be having second thoughts because he's leading North Korea into an uncertain future.
"There are certain conditions we want, and I think we'll get those conditions," Trump said, though he did not offer details. "If we don't, we won't" hold the summit, he said.
Trump added that "North Korea has a chance to be a great country. It can't be a great country under the circumstances they're living in right now. I think they should seize the opportunity, and we'll soon find out whether or not they want to do that."
The President also attempted to reassure Kim that he would remain in power under a deal to relinquish North Korea's nuclear weapons.
"I will guarantee his safety, yes," Trump said. "He will be safe, he will be happy, his country will be rich, his country will be hard-working and prosperous."
If Kim strikes a deal, Trump added: "He will be able to look back and be very proud of what he did for North Korea and, actually, for the world."
Moon, who met with Kim last month at the demilitarised zone between South Korea and North Korea, is expected to encourage Trump to follow through. South Korea's national security adviser, Chung Eui Yong, told reporters travelling with Moon from Seoul that the US-North Korea summit is still likely to happen.
"We believe there is a 99.9 per cent chance the North Korea-US summit will be held as scheduled," he said. "But we're just preparing for many different possibilities."
Sitting next to Trump in the Oval Office, Moon praised the President for his efforts on North Korea, saying Trump "has been able to achieve this dramatic change" in Kim's willingness to engage with foreign leaders. Over the past two months, the North Korean leader has made three trips outside this country - two to China and one over the border to South Korea - for the first time since taking power in 2011.
As he has before, Trump suggested that Pyongyang's shift to a more belligerent tone over the past week was the result of Kim's second meeting with Chinese President Xi Jinping. Beijing remains intent on ensuring that a nuclear deal would not draw North Korea closer to the United States and away from China's influence.
Moon said he has "every confidence" that Trump could reach a deal with Kim that would formally end the Korean War and bring "peace and prosperity" to North Korea.
"It's been a relationship that seems to be working," Trump said of his relationship with Kim, though he declined to say whether he has spoken directly with the North Korean leader.
Trump said if his summit with Kim moves forward "that will be a great, great thing for North Korea. If it doesn't, that's okay, too. Whatever it is, it is."
